mohatta2015.pdf

THE MATH 45
We can interpret f(t) as the correlation function for colored noise and g(t) = f*(—t)
as the matched filter in colored noise response (also called the generalized matched
filter response).
For the TDM case, the channel response for s(m) is h(t — mT), so that
/
oo
-OO
C~\h - τηΓ,ί)/ι(ί! - mT) dh. (2.68)
Unlike the AWGN case, f m (t) does not necessarily equal /n(i-mT), which is given
by
/
OO
-OO
/ O O
-oo
C-^tut-mT^i^dh
C~ l {h - mT, t - mT)h{h - mT) dh, (2.69)
unless it happens that C~ l {t\ — mT,t — mT) — C~ l {t\ — mT,t) for all m (i.e.,
C~ l {t\,t2) is periodic in ti with period T). Thus, we can no longer filter with a
common filter response and simply sample at different times.
2.3.10 Performance results
Simulation was used to generate results for both the matched filter and the matched
filter bound. General notes on simulation can be found in Appendix A.
Matched filter results were generated using 20 realizations of 1000 symbols each. 3
Simulation results for the matched filter bound were generated using 20,000 realizations
of 1 symbol each. Reference results are provided using (A.8).
Results were generated for QPSK and root-raised-cosine pulse shaping (rolloff
0.22). First consider the TwoTS channel defined in Chapter 1 (two symbol-spaced
paths with relative powers 0 and —1 dB and angles 0 and 90 degrees). BER vs.
Eb/N 0 is shown in Fig. 2.4. Observe that the matched filter experiences a "floor"
in that performance stops getting better with higher ΕΙ,/NQ towards the right side
of the plot.
By contrast, the matched filter bound shows no flooring. In fact, it agrees with
the reference result for an AWGN channel. This is because the matched filter
collects all the signal energy and ISI is perfectly removed (set to zero in this case).
Next consider the TwoFS channel (two fractionally spaced paths with relative
powers 0 and —1 dB and angles 0 and 90 degrees). Also consider two variations, in
which the angle of the second path is 0 degrees or 180 degrees. Simulated matched
filter bound results are shown in Fig. 2.5. Observe that performance depends
on the angle of the second path. This is because the two images of a particular
symbol are no longer orthogonal, but interact either constructively (second angle
= 0 degrees) or destructively (second angle = 180 degrees) or neither (second angle
= 90 degrees).
In general, we would like to plot BER vs. received Ε^,/Νο (not transmitted Ef,/N n ).
For the two-tap channel, by choosing the path angles to be 90 degrees apart, we
3 Slightly more than 1000 symbols were generated so that the middle 1000 symbols experienced
the same level of ISI.